diff --git a/Probability.md b/Probability.md index 4c7e544..1995a3d 100644 --- a/Probability.md +++ b/Probability.md @@ -2,9 +2,9 @@ ![](https://latex.codecogs.com/gif.latex?\binom{n}{n_{1},&space;n_{2},&space;..,&space;n_{m}}&space;=\frac{n!}{n_{1}!n_{2}!...n_{m}!}) - Phân phối n người khác nhau về m nhóm khác nhau với số lượng tương ứng: n1,n2,...,nm + Distribute n people to m group with respectively sizes: n1,n2...,nm - Ví du: abcd => 2 nhóm mỗi nhóm 2 người + Example: 4 people a,b,c,d => 2 groups, each group has 2 people: - (ab)(cd) - (ac)(bd) - (ad)(bc) @@ -12,6 +12,50 @@ - (bd)(ac) - (cd)(ab) - Kết quả: - + Result: + ![](https://latex.codecogs.com/gif.latex?\frac{4!}{2!2!}=6) + +* Number of vector (x1,x2..,xm) that: + - ![](https://latex.codecogs.com/gif.latex?x_{i}&space;>=0) + - ![](https://latex.codecogs.com/gif.latex?x_{1}+x_{2}+...+x_{m}=n) + + is: + + ![](https://latex.codecogs.com/gif.latex?\binom{n+m-1}{m-1}) + + If ![](https://latex.codecogs.com/gif.latex?x_{i}&space;>=1): + + ![](https://latex.codecogs.com/gif.latex?\binom{n-1}{m-1}) + + Example: If 8 identical blackboards are to be divided among +4 schools, how many divisions are possible?: + + x1+x2+x3+x4=8 + + ![](https://latex.codecogs.com/gif.latex?=\binom{8+4-1}{4-1}) + + if each school has to receive at least 1 blackboards: + + ![](https://latex.codecogs.com/gif.latex?=\binom{8-1}{4-1}) + +* Number of vector (x1,x2..,xm) that: + - ![](https://latex.codecogs.com/gif.latex?x_{i}&space;>=0) + - ![](https://latex.codecogs.com/gif.latex?x_{1}+x_{2}+...+x_{m}=n) + - ![](https://latex.codecogs.com/gif.latex?x_{i}&space;>=&space;min_{i}&space;,&space;i=1..m) + + ![](https://latex.codecogs.com/gif.latex?\binom{n'+m-1}{m-1}) + + with + + ![](https://latex.codecogs.com/gif.latex?n'&space;=&space;n&space;-&space;\sum_{i=1}^{m}min_{i}) + +* Binomial coefficient: + + ![](https://latex.codecogs.com/gif.latex?\binom{n}{k}&space;=&space;\frac{n!}{(n-k)!k!}) + + ![](https://latex.codecogs.com/gif.latex?\binom{n}{k}&space;=&space;\binom{n-1}{k-1}&space;+&space;\binom{n-1}{k}) + + ![](https://latex.codecogs.com/gif.latex?(x+y)^{n}&space;=&space;\sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}) + +*